Shafting misalignment fault diagnosis by means of motor speed signal and SVD-HT method

YU Zhen, AN Qi, SUO Shuangfu, QIU Zurong

(1. State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China；2. Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China)

Abstract： Aiming at the deficiency of diagnosis method based on vibration signal, a novel method based on speed signal with singular value decomposition and Hilbert transform (SVD-HT) is proposed. The fault diagnosis mechanism based on the speed signal is obtained by constructing the shaft misalignment fault model firstly. Then the SVD-HT method is applied to the processing of the speed signal. The accuracy of the SVD-HT method is verified by comparing the diagnosis results of the order spectrum method and the SVD-HT method. After that, the diagnosis results based on vibration signal and speed signal under no-load and load patterns are compared. Under the no-load pattern, the amplitudes of the speed signal components fr, 2fr and 4fr are linear with the misalignment. In addition, under the load pattern, the amplitudes of the speed signal components fr, 2fr and 4fr have a linear relationship with the load. However, the diagnosis result of the vibration signal does not have the above characteristics. The comparison results verify the robustness and reliability of the speed signal and SVD-HT method. The method presented in this paper provides a novel way for misalignment fault diagnosis.

Key words： servo motor； speed signal； misalignment fault； sigular value decomposition (SVD)； Hilbert transform (HT)

0 Introduction

Misalignment fault will aggravate the vibration of the motor drive system, which will cause the problems such as drive shaft bending, base looseness and bearing fault[1]. Moreover, the misalignment fault directly affects the transmission accuracy of precision instruments, and even causes engineering accidents on some high-speed occasions[2]. Therefore, the misalignment fault diagnosis of the motor drive shaft system plays an essential role in ensuring the transmission accuracy of precision instruments, extending the service life of the equipment, and ensuring the safety of production operations[3-5].

Misalignment fault diagnosis based on vibration signal has always been a hot topic for scholars[6-8]. Wang et al.[9]proposed an information interval spectrum (IIS) fault diagnosis method to improve the diagnosis efficiency in a robust noise environment. Wang et al.[10]focused on sensitive feature extraction and pattern recognition of rolling bearing fault diagnosis, and proposed an intelligent fault diagnosis method based on generalized composite multi-scale weighted permutation entropy, supervised Isomap (S-Iso), and marine predator algorithm Machine. Tavasolipour et al.[11]discussed the problem of fault estimation for nonlinear systems with Lipschitz nonlinearities. Su et al.[12]proposed a hybrid method for rolling bearing fault diagnosis based on simulated annealing particle swarm optimization and an improved kernel-based extreme learning machine. Xu et al.[13]developed an improved multi-scale convolutional neural network (CNN) that integrates the feature attention mechanism model to solve the poor performance of traditional CNN-based models in unstable and complex working environments. Kim et al.[14]outlined a domain adaptive semantic clustering method to diagnose faults in rotating machinery. In addition, because the noise in the signal affects the diagnosis and the separation of composite faults, Meng et al.[15]proposed a method combining periodic weighted kurtosis-sparse denoising and periodic filtering to extract repeated pulses of composite faults. Gonçalves et al.[16]proposed a pure output fault diagnosis method based on Markov parameters using random implementation theory. Chen et al.[17]proposed a fault diagnosis method for the long-term safe operation of rotating machinery based on improved composite multi-scale fuzzy entropy, topology learning, and out-of-sample embedding and support vector machine based on ocean predator algorithm. Zhang et al.[18]proposed a method based on hybrid attention improved residual network to diagnose faults by highlighting the fault characteristics of the fundamental frequency band of wavelet coefficients and convolution channel.

Although the diagnosis method based on vibration signal has achieved remarkable results, with the deepening of research, the drawbacks of this method gradually appear. First of all, additional errors may be introduced during the acquisition and processing of vibration signals, reducing the signal-to-noise ratio of the weak fault information in the original signal, which will lead to the failure of diagnosis[1-2]. Secondly, due to the speed fluctuation of the shafting, the collected signal is usually non-stationary, or second-order cyclostationary and its characteristic parameters change with time[19-20]. Moreover, the installation position of the sensor largely determines the quality of the collected signal[21-22]. In addition, additional costs, environmental influences, and other factors also limit the development of vibration signal diagnosis methods[1].

In recent years, electrical methods have gradually been pursued by scholars for their “non-intrusive” advantages. The electrical method realizes the system fault diagnosis by collecting signals such as phase current, electromagnetic torque, or speed and cooperating with signal processing technology[23]. Antonino-Daviu et al.[24]diagnosed and identified the misalignment fault based on the starting current signal of the motor. The research results show that using the motor current signal to realize mechanical fault diagnosis and monitoring has a good application prospect. Verucchi et al.[2]found that current and load torque signals have different diagnostic effects on angular misalignment fault under different types of couplings and change with the change of load. Detected the coupling misalignment based on torque signal, and the effectiveness of torque signal in misalignment fault diagnosis was verified. Yang et al.[25]carried out fault diagnosis of motor bearing based on the characteristic analysis method of speed signal, which preliminarily showed the feasibility of the characteristic analysis method of the speed signal for mechanical fault diagnosis.

At present, the misalignment fault diagnosis of the electrical method is mostly based on motor current signal. However, the acquisition of the current signal will be disturbed by the sampling bias of the embedded system, current fundamental frequency component, and harmonics[1]. In comparison, collecting the speed signal through the encoder has the advantages of convenient acquisition, higher sampling accuracy, and lower cost. At the same time, the diagnosis method based on the speed signal can realize real-time status monitoring[25].

Based on the above analysis, the misalignment fault diagnosis based on the speed signal has a broader application prospect. Therefore, the research on shafting misalignment fault diagnosis based on speed signal was carried out in this study. Combined with singular value decomposition and Hilbert transform (SVD-HT) method, a novel misalignment fault diagnosis method was proposed.

1 Misalignment fault model and diagnosis mechanism

Misalignment mainly includes three types： parallel misalignment, angular misalignment, and comprehensive misalignment. Considering that comprehensive misalignment is a combination of parallel misalignment and angular misalignment, the following two types will be described in detail.

1.1 Parallel misalignment fault

Parallel misalignment fault refers to the phenomenon that the drive motor shaft and the shaft system are parallel to each other but do not overlap[25]. The parallel misalignment fault model is shown in Fig.1.

Fig.1 Parallel misalignment fault model

In Fig.1,Mis the center of the motor shaft, andNis the center of the drive shaft. The misalignment between the center of the motor shaft and the drive shaft indicates that there is a misalignment fault.θis the angular position of the drive shaft, anddis the displacement of parallel misalignment.

Assuming that the speed of the transmission system is stable, the following formula is satisfied

θ=ωmt+φ,

(1)

whereθis the angular position of the drive shaft,ωmis the rotational angular velocity of the drive shaft,tis the time, andφis the initial angular position.

The expression of the motor drive torqueTmcan be obtained as[27]

(2)

wheremris the equivalent mass of the rotor system,gis gravitational acceleration,xis the displacement of the motor axis in thex-direction；yis the displacement of the motor axis in they-direction；TLis the load torque of the rotor system.

It can be seen from the driving torque Eq.(2) that when there is a parallel misalignment fault in the system, the electromagnetic torque of the motor must not only meet the requirements of the load torqueTL, but also overcome the fluctuating torqueTCintroduced by the fault. The equation ofTCis

(3)

In order to analyze the frequency components in fluctuating torque, combined with Eq.(1), Eq.(3) can be simplified to

TC=Acos(ωat+φ),

(4)

whereωa=2πfr,fris the rotation frequency； andAis the constant coefficient. It can be seen that the fluctuating torqueTCis a sinusoidal signal of 1 time the frequencyfr.

Based on the above analysis, the diagnosis mechanism of parallel misalignment can be obtained： parallel misalignment fault will introduce a fluctuating torque of 1 time the rotation frequency into the load torque, and the electromagnetic torque will produce a fluctuating torque against it. The electromagnetic torque fluctuation will introduce the speed fluctuation of the same frequency into the motor speed signal, that is, the amplitude of 1 time of the rotation frequencyfrwill increase in the frequency spectrum analysis of the speed signal.

1.2 Angular misalignment fault

Angular misalignment fault refers to that under the premise that the drive shaft system axis and the axis of the motor shaft intersect at one point but are not parallel to each other, and there is a certain angle between the axis of the motor shaft and the axis of the drive shaft[25]. The angular misalignment fault model is shown in Fig.2.

Fig.2 Angular misalignment fault model

In order to study the influence of angular misalignment on motor operation, the additional torque caused by angular misalignment is analyzed firstly.

As shown in Fig.2, the deflection angle formed between the motor shaft and the rotating shaft isα. Thez-axis coincides with the transmission shaft axis, and the motor torqueTecan be decomposed into the torque componentTzdriving the rotating shaft along thez-axis and the bending moment componentTsperpendicular to the rotor shaft. There exists the relationship[27]as

(5)

For the motor rotor system with the deflection angleα, the angular velocities of the two satisfy[27].

(6)

Carrying out the Taylor expansion of Eq.(6), we can get

(7)

where

(8)

After differential processing to Eq.(7), we have

(9)

(10)

In the actual system, since the torsional vibration amplitude of high-frequency multiplier is weak and can be approximately ignored, the sinusoidal component of double-conversion frequency is mainly considered, so Eq.(10) can be further simplified as

Ts=JrωmA2sin(ωmt+φm)+A0-

A2cos(ωmt+φm),

(11)

whereωm=2πfr, andfris the rotation frequency. It can be seen that the fluctuating torqueTsis a sinusoidal signal of 1 time the frequencyfr.

Based on the above analysis, the diagnosis mechanism of angular misalignment fault can be obtained： angular misalignment fault will introduce a fluctuating torque of 1 time the rotation frequency into the load torque, and the electromagnetic torque will produce a fluctuating torque against it. The electromagnetic torque fluctuation will introduce the speed fluctuation of the same frequency into the motor speed signal. That is, the amplitude of 1 time of the rotation frequencyfrwill increase in the frequency spectrum analysis of the speed signal.

2Experimental test

2.1 Test rig

The overall structure diagram and physical drawings of the test rig is shown in Fig.3.

(a) Overall structure diagram

(b) Physical drawingsFig.3 Test rig

The test rig consists of 8 parts： servo motor, coupling, dynamometer, misalignment adjustment device, speed test module, vibration test module, data acquisition module, and host computer.

The servo motor is an SMA13-46P1B servo motor produced by Monde Electric Co., Ltd. The rated speed of the motor is 2 000 r·min-1, and the rated torque of the motor is 11 N·m. The speed accuracy of the motor is 0.1 r·min-1, and the speed range of the motor is 0-10 400 r·min-1. The coupling is an Oldham coupling, with an allowable radial misalignment of 2 mm and an allowable angular misalignment of 3°. The eddy current dynamometer is used to provide load, and the rated torque of the dynamometer is 10 N·m. The misalignment adjustment device is shown in Fig.3, which consists of 4 sets of cages and adjustment knobs. When adjusting the parallel misalignment, the adjustment knobs on the same side rotate simultaneously, resulting in parallel misalignment. When adjusting the angular misalignment, we adjust a single knob on the same side to produce angular misalignment. The vibration test module includes two accelerometers perpendicular to each other. The vibration signal is sampled at equiangular spacing. The accelerometer is a precision quartz shear integrated circuit piezoelectricity (ICP) type with a sensitivity of 1 000 mV/g and a frequency range of 0.05 kHz-25 kHz. In each measurement, the output of the two acceleration sensors will be recorded simultaneously. The speed test module is realized by the angle encoder of the motor, and the speed signal also adopts the equiangular spacing sampling method. The data acquisition module adopts PXIe7961 acquisition system produced by NI-Company, which is used to collect the signals of angle encoder and accelerometers. The speed signal acquisition accuracy of the PXIe acquisition system is 0.1 r·min-1. And the PXIe acquisition system can realize sampling with variable sampling frequency from 0 Hz-160 MHz. For different speeds, the speed signal and the vibration signal are collected at different sampling rates to ensure that the signal is sampled once per degree. Thus the sampling rate of the acquisition system is always 360 times of the speed.

2.2 Experimental process

The whole experiment can be divided into two parts： no-load pattern and load pattern. The experimental steps of each part can be divided into the following three steps： Firstly, the initial position adjustment of the test rig; And then the driving and loading parameter settings; Finally, the parameter settings for the misalignment experiment.

Before adjusting the initial position of the test rig, we connect the components of the test rig described in section 2.1 in order, and adjust the initial position of the test rig based on the double dial indicator method. Two TESA probes are used to adjust the coaxiality error between the motor output shaft and the dynamometer input shaft to within 20 μm, which is used as a parameter setting under the alignment condition.

After the adjustment of the test rig, the driving and loading parameters need to be set. In order to verify the accuracy and robustness of the diagnosis method in this study, two experiments of no-load pattern and load pattern are set, respectively. In the no-load pattern, the dynamometer is adjusted to no-load mode (0 N·m). When adjusting the drive motor to speed mode, the motor speed gradually increases from 0 r·min-1to 1 200 r·min-1, and the speed increment is 300 r·min-1. The parameter setting of the drive motor in load pattern is the same as that in no-load pattern. But the dynamometer is adjusted to torque loading mode, the torque range is 0-10 N·m, and the torque increment is 12.5%FS.

After completing the above adjustment and parameter setting, the next step is to set the misalignment parameters for the experiment. Parameters adjustment of parallel misalignment test： adjust the parallel misalignment parameters based on is to the double dial indicator method: Use the misalignment adjustment device on the test rig, rotate the adjustment knob on the same side simultaneously, and set the adjustment amount of the coaxiality error between the motor output shaft and the dynamometer input shaft to 1 mm and 2 mm, respectively. Parameters adjustment of angular misalignment test is to adjust the parallel misalignment parameters based on the double dial indicator method: Use the misalignment adjustment device on the experimental bench, rotate the single adjustment knob on the same side, and set the adjustment amount of the coaxiality error between the motor output shaft and the dynamometer input shaft to 1.64 mm and 3.28 mm, respectively, which correspond to angular misalignment of 1° and 2°, respectively.

The equiangular spacing sampling method is used in this study. In order to ensure the reliability of the experimental results, the speed of multiple sampling points in each cycle is measured three times under each working condition, and the three measurement results are averaged to obtain the speed of each sampling point.

2.3 Signal processing method

The order spectrum analysis of the speed signal can be used to obtain the frequency spectrum of the speed signal. However, the order spectrum of the rotating signal may contain periodic noise signals, so the fault signal does not dominate in the frequency spectrum, thereby interfering with the fault diagnosis. In order to realize the fault diagnosis more accurately, the SVD-HT method is applied to the data processing to obtain the characteristics of the speed signal under different misalignment fault states.

2.3.1 Signal processing principle of SVD

For data processing, it is first necessary to remove the DC component in the speed signal. The SVD filtering stage is included in the data processing of removing the DC component of the speed signal.

The SVD filtering algorithm uses the Hankel matrix structure to decompose the speed signal into a series of signal subspaces[28]. Assuming that the speed signalX={x1,x2,…,xN}, where 1, 2,…,Nare sampling points, the speed signal can be written in the matrix form as

(12)

In the above equation, the range ofnis 1

(13)

Eq.(13) can also be expressed as

(14)

whereris the rank,σiis the singular values or weights withσ1≥σ2≥…≥σr>0, and eachDi(orσi) corresponds to singular vectorsuiandvi.

Singular valuesσ1,σ2, …,σrare responses to singular values of different frequency components. The singular value can be used to obtain a reasonable, effective rank of various frequency components. The maximum singular value corresponds to the DC component, and the smaller singular value corresponds to the fluctuation signal caused by coupling misalignment fault.

Therefore, after the effective rank of the singular matrix is determined, the singular value with the maximum effective rank is eliminated. Then through the inverse operation of SVD, the matrix estimation of the DC component is obtained. Finally, through the inverse reconstruction of the phase space, the speed signal component caused by the coupling misalignment fault can be obtained. The reconstructed speed signal componentsxi(k)(k=1,2,…,m) or (k=1,2,…,n) corresponding to orderiare obtained from the column or row vectors ofσiDi.

2.3.2 Signal processing principle of HT

(15)

Then the analytical signals of the speed signal component can be constructed as

(16)

The amplitude functionφ(t) of the speed signal componentx(t) can be obtained by

(17)

The phase functionφ(t) of the speed signal componentx(t) can be obtained by

(18)

Then the instantaneous frequency of the speed signal componentx(t) can be calculated as

(19)

After the process above, the Hilbert transform of each component can be expressed as

(20)

where Re represents the real part；nis the number of eigenmode functions；Ai(t) is theith component of the speed signal caused by the coupling misalignment fault； andωi(t) is the instantaneous frequency of theith component.

After HT processing, the variation law of the amplitude of speed signal components with time and frequency in the whole frequency range can be accurately described. Combined with the frequency analysis of the speed signal component generated by the misalignment fault, the type and value of misalignment fault parameters can be accurately diagnosed. It means that the type and value of misalignment fault parameters can be accurately diagnosed through the amplitude of 1 time of the rotation frequencyfrand other components in the SVD-HT spectrum of the speed signal. The scheme of signal processing is shown in Fig.4.

Fig.4 Scheme of signal processing

3 Comparison results and analysis

The comparison results can be divided into three parts： Firstly, the diagnosis results of order spectrum analysis and the SVD-HT method are compared to verify the accuracy of the SVD-HT signal processing algorithm. Secondly, the diagnosis results of vibration signal and speed signal under no-load pattern are compared to verify the superiority of speed signal diagnosis method. Finally, the diagnosis results of vibration signal and speed signal under load pattern are compared to verify the robustness and reliability of speed signal and SVD-HT diagnosis method.

3.1 Diagnosis results of different signal processing algorithms

In order to verify the accuracy of SVD-HT method, order spectrum analysis and SVD-HT method are used to process the speed signal under 1 mm parallel misalignment at 300 r·min-1. Fig.5 shows the comparison results of speed signals under 1 mm parallel misalignment at 300 r·min-1.

(a) Order spectrum analysis

(b) SVD-HT methodFig.5 Comparison results of speed signals under 1 mm parallel misalignment at 300 r·min-1

Fig.5(a) shows the diagnosis results of order spectrum analysis. It can be seen that the order spectrum of the speed signal contains more periodic noise signals. The actual fault signal is not dominant in the spectrum, which leads to the error of diagnosis results. Fig.5(b) shows the diagnosis results of the SVD-HT method. After SVD-HT processing, the noise signal in the signal spectrum is obviously suppressed. The fault signal and the fault frequency multiplier signal are consistent with the experimental setting parameters. The comparison result verifies the accuracy of the SVD-HT signal processing algorithm.

3.2 Diagnosis results under no-load pattern

In order to verify the superiority of the speed signal diagnosis method, the diagnosis results of vibration signal and speed signal under a no-load pattern are compared and analyzed. The experimental parameters of the no-load pattern are described in Section 2.2： the output torque of the dynamometer is 0 N·m. The speed of the drive motor increases from 0 r·min-1to 1 200 r·min-1, and the speed increment is 300 r·min-1. The parallel misalignment parameters are 0 mm, 1 mm, and 2 mm, respectively, and the angular misalignment parameters are 0°, 1°, and 2°, respectively.

In the comparison results, the processing results of vibration signals are based on order spectrum analysis and power spectrum analysis. By analyzing the vibration signals in thexandydirections of the shafting system, the axis trajectory of the shafting system under different misalignment conditions can be obtained. In contrast, the processing result of the speed signal is based on the SVD-HT method.

3.2.1 Parallel misalignment fault diagnosis results

Figs.6 and 7 are the order spectrum and power spectrum of the vibration signal in thexandydirections under the alignment condition, respectively. Fig.8 shows the axis trajectory under alignment conditions.

It can be seen from Figs.6 and 7 that under the alignment condition, there are four componentsfr, 2fr, 4frand 8frin the vibration signal spectrum. At each speed, the amplitudes of the four components in thexandydirections are almost the same. And the power values offrcomponent in thexandydirections are almost the same, while the power values of 2fr, 4frand 8frcomponents are zero. Under this condition, the axis trajectory is circular.

Fig.6 Order spectrum and power spectrum of vibration signal inx-direction under alignment condition

Fig.7 Order spectrum and power spectrum of vibration signal in y direction under alignment condition

Fig.8 Axis trajectory under alignment condition

Figs.9 and 10 show the order spectrum and power spectrum of vibration signal in thexandydirections under the condition of 1 mm parallel misalignment, respectively. Fig.11 shows the axis trajectory under the condition of 1 mm parallel misalignment.

Combining Figs.9 and 10, under the condition of 1 mm parallel misalignment, there are four componentsfr, 2fr, 4fr, and 8frin the vibration signal spectrum. At each speed, the amplitude and power of componentsfr, 4fr, and 8frin thexandydirections are almost the same. However, with the increase of speed, the amplitude and power of component 2frsometimes increase and sometimes decrease, and there is no obvious change trend. Moreover, as shown in Fig.11, the axis trajectory changes with the change of speed, and the trajectory gradually changes from a crescent shape to the shape of number 8.

Fig.9 Order spectrum and power spectrum of vibration signal in x-direction under the condition of 1 mm parallel misalignment

Fig.10 Order spectrum and power spectrum of vibration signal in y-direction under the condition of 1 mm parallel misalignment

Fig.11 Axis trajectory under the condition of 1 mm parallel misalignment

Figs.12 and 13 show the order spectrum and power spectrum of vibration signal in thexandydirections under the condition of 2 mm parallel misalignment, respectively. Fig.14 shows the axis trajectory under the condition of 2 mm parallel misalignment.

Combining Figs.12 and 13, under the condition of 2 mm parallel misalignment, there are four componentsfr, 2fr, 4fr, and 8frin the vibration signal spectrum. At each speed, the amplitude and power of componentsfr, 4fr, and 8frin thexandydirections are almost the same. However, with the increase of speed, the amplitude and power of component 2frsometimes increase and sometimes decrease, and there is no obvious change trend. Moreover, as shown in Fig.14, the axis trajectory changes with the change of speed, and the trajectory gradually changes from the shape of the standard number 8 to the shape of the specific number 8.

Fig.12 Order spectrum and power spectrum of vibration signal in x-direction under the condition of 2 mm parallel misalignment

Fig.13 Order spectrum and power spectrum of vibration signal in y direction under the condition of 2 mm parallel misalignment

Fig.14 Axis trajectory under the condition of 2 mm parallel misalignment

Fig.15 shows the diagnosis results based on speed signal and SVD-HT method under alignment conditions.

As shown in Fig.15, under the alignment condition, there are four componentsfr, 2fr, 4fr, and 8frin the speed signal spectrum. These components may be caused by cogging torque and electromagnetic torque ripple. It is worth noting that the amplitudes of the four components are almost the same at each speed.

Fig.15 Diagnosis results based on speed signal and SVD-HT method under alignment conditions

Figs.16 and 17 are the diagnosis results based on the speed signal and the SVD-HT method under 1 mm and 2 mm parallel misalignment conditions, respectively.

Fig.16 Diagnosis results based on speed signal and SVD-HT method under the condition of 1 mm parallel misalignment

Fig.17 Diagnosis results based on speed signal and SVD-HT method under the condition of 2 mm parallel misalignment

In combination with Figs.16 and 17, the amplitude of componentfris greater than that of components 2fr, 4fr, and 8fr. Excitingly, the amplitude of componentfrgradually increases with the increase of parallel misalignment parameters. In addition, compared with Fig.15, it can be seen that the amplitudes of components 2fr, 4fr, and 8fr. under misalignment conditions are greater than those under alignment conditions. This phenomenon corresponds to the diagnosis mechanism in section 1.1 that the electromagnetic torque gradually increases with the increase of the misalignment parameters. The increase of electromagnetic torque will cause the amplitude of the componentfrto increase. It can be concluded that the amplitude of componentfrcan be used for parallel misalignment fault diagnosis.

Beside the experiments above,some more experiments have been done for 0.25 mm, 0.5 mm, 0.75 mm, 1.25 mm, 1.5 mm and 1.75 mm to find the relationship between the amplitude of different components and the parallel misalignment parameters. The amplitude variation results of the speed signal componentsfr, 2fr, and 4frunder the total nine groups of misalignment parameters can be obtained, as shown in Fig.18. The nine groups of parallel misalignment parameters are 0 mm, 0.25 mm, 0.5 mm, 0.75 mm, 1 mm, 1.25 mm, 1.5 mm, 1.75 mm, and 2 mm, respectively.

Fig.18 Amplitude variation results of the speed signal components fr, 2fr, and 4fr, under nine groups of parallel misalignment parameters

According to Fig.18, the amplitudes of the speed signal componentsfr, 2fr, and 4frgradually increase with the increase of the misalignment parameters, and the relationship between the two is linear. With the increase of rotating speed, the increase rate offramplitude is more obvious than other components. At 1 200 r·min-1, the amplitude offrincreased by more than 200%. Therefore, the parallel misalignment fault can be effectively diagnosed by the amplitude of componentfr.

3.2.2 Angular misalignment fault diagnosis results

Figs.19 and 20 show the order spectrum and power spectrum of vibration signal in thexandydirections under the condition of 1° angular misalignment, respectively. Fig.21 shows the axis trajectory under the condition of 1° angular misalignment.

Fig.19 Order spectrum and power spectrum of vibration signal in x-direction under the condition of 1° angular misalignment

Fig.20 Order spectrum and power spectrum of vibration signal in y-direction under the condition of 1° angular misalignment

Combining Figs.19 and 20, there are four componentsfr, 2fr, 4fr, and 8frin the vibration signal spectrum. At each speed, the amplitude and power of components 2fr, 4fr, and 8frinxandydirections are almost the same. With the increase of speed, the amplitude and power of the componentfrin thex-direction increase significantly. However, in they-direction, the amplitude and power of the componentfrat different speeds are almost the same. Moreover, as shown in Fig.21, the axis trajectory is elliptical, and the axis trajectory gradually changes with the change of speed.

Figs.22 and 23 show the order spectrum and power spectrum of vibration signal in thexandydirections under the condition of 2° angular misalignment, respectively.

Fig.24 shows the axis trajectory under the condition of 2° angular misalignment.

Fig.24 Axis trajectory under the condition of 2° angular misalignment

Combining Figs.22 and 23, there are four componentsfr, 2fr, 4fr, and 8frin the vibration signal spectrum. At each speed, the amplitude and power of components 2fr, 4fr, and 8frinxandydirections are almost the same. With the increase of speed, the amplitude and power of the componentfrin thex-direction increase significantly. However, in they-direction, the amplitude and power of the componentfrat different speeds are almost the same. Moreover, as shown in Fig.24, the axis trajectory is elliptical, and the axis trajectory gradually changes with the change of speed.

The above contents are the diagnosis results based on the vibration signal, and the diagnosis results based on the speed signal are introduced below. Because there are too many spectrum diagrams to show one by one, so they will not be shown here. From the diagnosis results of parallel misalignment, there are mainlyfr, 2fr, and 4frcomponents in the SVD-HT spectrum. According to the processing flow of speed signal in section 2.3.2, the amplitude change results of speed signal componentsfr, 2fr, and 4frare directly taken for the summary of the angular misalignment results. In order to find the relationship between the amplitude of different components and the angular misalignment parameters, some experiments have been done for 0°, 0.25°, 0.5°, 0.75°, 1°, 1.25°, 1.5°, 1.75°, and 2°. The amplitude variation results of the speed signal componentsfr, 2fr, and 4frunder the total nine groups of misalignment parameters can be obtained, as shown in Fig.25.

Fig.25 Amplitude variation results of speed signal components fr, 2fr, and 4fr, under nine groups of parallel misalignment parameters

According to Fig.25, the amplitudes of the speed signal componentsfr, 2fr, and 4frgradually increase with the increase of the misalignment parameters, and the relationship between the two is linear. With the increase of rotating speed, the increase rate offramplitude is more obvious than other components. At 1 200 r·min-1, the amplitude offrincreased by more than 200%. Therefore, the angular misalignment fault can be effectively diagnosed by the amplitude of componentfr.

The above contents are the diagnosis resultsof vibration signal and speed signal under a no-load pattern. Comprehensive comparison results show that the characteristic parameters of vibration signal change with the change of speed. When the shaft speed fluctuates, the collected vibration signal is non-stationary, and its characteristic parameters will change with speed change. Especially when diagnosing parallel misalignment faults, the amplitude and power of component 2frsometimes increase and sometimes decrease, and there is no obvious change trend. Therefore, the diagnosis method based on vibration signal may lead to erroneous results.

In contrast, since only the force along the circumferential direction can affect the speed signal, the SVD-HT results of the speed signal are stable. The amplitudes of the speed signal componentsfr, 2frand 4frgradually increase with the increase of the misalignment parameters, and the relationship between the two is linear. The amplitude of componentfrcan effectively diagnose the type and value of misalignment fault parameters. The comparison results verify the superiority of the speed signal diagnosis method.

3.3 Diagnosis results under load pattern

The amplitude of the signal component will vary with the load. Therefore, in order to verify the robustness and reliability of speed signal and SVD-HT diagnosis method, the diagnosis results of vibration signal and speed signal under load pattern are compared and analyzed.The experimental parameters of the load pattern are described in Section 2.2： The dynamometer is adjusted to torque loading mode, the torque range is 0-10 N·m, and the torque increment is 12.5%FS. The motor speed gradually increases from 0 r·min-1to 1 200 r·min-1, and the speed increment is 300 r·min-1. The parallel misalignment parameters are 0 mm, 1 mm, and 2 mm, respectively, and the angular misalignment parameters are 0°, 1°, and 2°, respectively. Because there are too many spectrum diagrams to show one by one, they will not be shown here. According to the processing flow of speed signal in section 2.3.2, the amplitude change results of speed signal componentsfr, 2fr, and 4frare directly taken for presentation.

3.3.1 Parallel misalignment fault diagnosis results

Figs.26- 28 show the amplitude variation results of the vibration signal componentsfr, 2fr, and 4frinxandydirections under three groups of misalignment parameters. The three groups of parallel misalignment parameters are 0 mm, 1 mm and 2 mm, respectively.

According to Figs.26-28, with the increase of the load, the amplitudes of the vibration signal componentsfr, 2fr, and 4frinxandydirections sometimes increase and sometimes decrease, and there is no obvious change trend. In addition, the results at each speed are irregular. Therefore, it is difficult to diagnose the misalignment fault by using a vibration signal.

Fig.26 Amplitude variation results of vibration signal components fr, 2fr and 4fr in x and y directions under alignment condition

Fig.27 Amplitude variation results of vibration signal components fr, 2fr, and 4fr in x and y directions under 1 mm parallel misalignment

Fig.28 Amplitude variation results of vibration signal components fr, 2fr, and 4fr in x and y directions under 2 mm parallel misalignment

Figs.29-31 show the amplitude variation results of the speed signal componentsfr, 2fr, and 4frunder three groups of misalignment parameters. The three groups of parallel misalignment parameters are 0 mm, 1 mm and 2 mm, respectively.

Fig.29 Amplitude variation results of speed signal components fr, 2fr and 4fr under alignment condition

Fig.30 Amplitude variation results of speed signal components fr, 2fr and 4fr under 1 mm parallel misalignment

Fig.31 Amplitude variation results of speed signal components fr, 2fr and 4fr under 2 mm parallel misalignment

According to Figs.29-31, under each speed, the amplitudes of the speed signal componentsfr, 2frand 4frgradually increase with the increase of the load, and the relationship between the two is linear. Among them, the increase rate offramplitude is more obvious than that of other components. When the rated load is reached, the amplitude offrincreases by more than 100%. Therefore, the parallel misalignment fault can be effectively diagnosed by the amplitude of componentfr.

3.3.2 Angular misalignment fault diagnosis results

Figs.32 and 33 show the amplitude variation results of the vibration signal componentsfr, 2frand 4frinxandydirections under two groups of misalignment parameters. The two groups of angular misalignment parameters are 1° and 2°, respectively.

Fig.32 Amplitude variation results of vibration signal components fr, 2fr and 4fr in x and y directions under 1° angular misalignment

Fig.33 Amplitude variation results of vibration signal components fr, 2fr and 4fr in x and y directions under 2° angular misalignment

Figs.34 and 35 show the amplitude variation results of the speed signal componentsfr, 2frand 4frunder two groups of angular misalignment parameters are 1° and 2°, respectively.

Fig.34 Amplitude variation results of speed signal components fr, 2fr and 4fr under 1° angular misalignment

Fig.35 Amplitude variation results of speed signal components fr, 2fr and 4fr under 2° angular misalignment

According to Figs.32-35, with the increase of the load, the amplitudes of the vibration signal componentsfr, 2frand 4frinxandydirections sometimes increase and sometimes decrease, and there is no obvious change trend. In addition, the results at each speed are irregular. In comparison, under each speed, the amplitudes of the speed signal componentsfr, 2frand 4frgradually increase with the increase of the load, and the relationship between the two is linear. Among them, the increase rate offramplitude is more obvious than that of other components. When the rated load is reached, the amplitude offrincreases by more than 100%.

The above contents are the diagnosis results of vibration signal and speed signal under load pattern. Comprehensive comparison results show that the characteristic parameters of vibration signal change with the change of load. When the shaft speed fluctuates, the collected vibration signal is non-stationary, and its characteristic parameters will change with speed change. With the increase of load, the amplitudes of the vibration signal componentsfr, 2frand 4frhave an upward trend, but there is no obvious law. Therefore, the vibration signal diagnosis method cannot quantitatively analyze the misalignment fault, leading to erroneous results.

In comparison,under each speed, the amplitudes of the speed signal componentsfr, 2frand 4frgradually increase with the increase of the load, and the relationship between the two is linear. Among them, the increase rate offramplitude is more obvious than that of other components. When the rated load is reached, the amplitude offrincreases by more than 100%. This rule can effectively realize the quantitative analysis of misalignment fault diagnosis. The comparison results verify the robustness and reliability of the speed signal and the SVD-HT diagnosis method.

Therobustness and reliability refers to the ability of the diagnosis system to correctly complete the fault diagnosis task in the presence of noise, interference, etc., while maintaining a low false alarm rate and under-reporting rate. The stronger the robustness and the reliability of the diagnosis method, the lower the false alarm rate and under-reporting rate are. Therefore, the evaluation index of robustness and reliability are false alarm rate and under-reporting rate. From the previous experiment about diagnosing the parallel and angular misalignment fault under different rotation speeds and different loads, it can be seen that there is a linear relationship between the amplitude offrand the value of misalignment fault under different rotation speeds and different loads. Therefore, the false alarm rate and under-reporting rate are 0%. In order to verify the robustness and reliability of the proposed method, three groups of independently repeating experiments of diagnosing the the parallel and angular misalignment fault have been done. The false alarm rate and under-reporting rate of the experiments are shown in Table 1.

Table False alarm rate and under-reporting rate of experiments

It can be seen from Table 1 that the false alarm rate and under-reporting rate are always 0%. Therefore, the robustness and reliability of the proposed method are pretty good.

4 Conclusions

This paper presents a novel method for shaft misalignment diagnosis. This method combines the motor speed signal and the SVD-HT method, which overcomes the shortcomings of the diagnosis method based on vibration signal. The fault diagnosis mechanism based on the speed signal is obtained by constructing the shaft misalignment fault model： the misalignment fault will increase the amplitude of the speed signal componentfr. The diagnosis results based on vibration and speed signals under no-load and load patterns are compared and analyzed. The comparison results verify the robustness and reliability of the speed signal and the SVD-HT diagnosis method.

In summary, the diagnosis method based on speed signal and SVD-HT proposed in this paper can effectively realize the analysis of thetype and value of misalignment fault parameters. However, the presented methods have higher requirements for the motor encoder’s sampling accuracy and sampling frequency. The fault diagnosis method based on the speed signal analysis under the under-sampling condition will be studied in the future.